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120                       ANALYTICAL MECHANICS
DISCUSSION. - Instead of considering the masses separately we can consider them as a single moving system and write a single force equation.
(total moving mass) X (acceleration) = sum of the forces, or                              (mi + ra2) -~ = m& - m2g,
which is identical with equation (c).
1. Two particles of mass mi and mz are suspended by a string which is slung over a smooth table. A third particle of mass ms is attached to that portion of the string which is on the table. Prove that when the system is left to itself it will move with an acceleration of
mi + m2
2.   In the preceding problem suppose the table to be rough and find "the acceleration,   p, = 0.5.
3.   Discuss Atwood's machine supposing a frictional force to act between the string and the pulley (the latter is supposed to be fixed) ; take the frictional force to be equal to the tensile force in that portion of the string which is moving up.
109.  Case II. Parabolic Motion, or Motion of Projectiles. 
Consider the motion of a particle which is projected in a direction making an angle a with the horizon. When we neglect the resistance of the air, the only force which acts upon the particle is its weight, mg. Taking the plane of motion to be the :n/-plane, Fig. 66, we have
/"7/v                     /77*/
where -j- and ~ are the components of the acceleration
along the axes.   Integrating equations (1) and (2) we get
i=ci, and                                 y= - gt + c2.